Broadcasts on Paths and Cycles
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A broadcast on a graph G=(V,E) is a function f: V⟶{0,...,diam(G)} such that f(v)≤ e_G(v) for every vertex v∈ V, wherediam(G) denotes the diameter of G and e_G(v) the eccentricity of v in G. The cost of such a broadcast is then the value ∑_v∈ Vf(v).Various types of broadcast functions on graphs have been considered in the literature, in relation with domination, irredundence, independence or packing, leading to the introduction of several broadcast numbers on graphs. In this paper, we determine these broadcast numbers for all paths and cycles, thus answering a question raised in [D. Ahmadi, G.H. Fricke, C. Schroeder, S.T. Hedetniemi and R.C. Laskar, Broadcast irredundance in graphs. Congr. Numer. 224 (2015), 17--31].
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